I'm happy to be corrected, but I don't think you can get into the BIOS without a CPU. However (if you had the flashback feature) then you can update the BIOS that way without a CPU.
To run a 13th gen CPU you will need BIOS 2004 or later (or the equivalents for your board). So, I think it's true to say that if the BIOS is older than 2004, then you will need a 12th gen CPU to access the BIOS and update it that way.
Z690 Hero, BIOS 3401, MEI 2406.5.5.0, ME Firmware 16.1.30.2361, 7000X Case, RM1000x PSU, i9 12900K, ASUS TUF OC 3090TI, 2 x 16GB Corsair RAM @ 5200MHz, Windows 11 Pro 23H2, Corsair H150i Elite AIO, 4x Corsair RGB fans, 3x M.2 NVME drives, 2x SATA SSDs, 2x SATA HDs.